where \(P^k(I_i)\)  represents all polynomials of degree at most \(k\)  on \(I_i\) . Then, the semi-discrete UWDG scheme proposed in [1]} is based on repeated integration by parts, and to find \(u^h \in V^k_h\) , such that for any test function \(v^h \in V^k_h\) , there holds the equality:
\( \nonumber & \int _{\Omega } (u^h)_t v^h dx = \int _{\Omega } f(u^h)v^h_x dx +\int _{\Omega } u^h v^h_{xxx} dx-\sum _{i=1}^{N_x} ( \hat{f}_{i+\frac{1}{2}} (v^h)^-_{i+\frac{1}{2}}- \hat{f}_{i-\frac{1}{2}} (v^h)^+_{i-\frac{1}{2}}) \\ \nonumber &- \sum _{i=1}^{N_x} (\hat{u}^h_{i+\frac{1}{2}} (v^h_{xx})^-_{i+\frac{1}{2}} -\hat{u}^h_{i-\frac{1}{2}} (v^h_{xx})^+_{i-\frac{1}{2}} )+\sum _{i=1}^{N_x} ( (\widetilde{u}^h_x)_{i+\frac{1}{2}}(v^h_x)^-_{i+\frac{1}{2}} - (\widetilde{u}^h_x)_{i-\frac{1}{2}}(v^h_x)^+_{i-\frac{1}{2}} ) \\&-\sum _{i=1}^{N_x} ( (\check{u}^h_{xx})_{i+\frac{1}{2}} (v^h)^-_{i+\frac{1}{2}} - (\check{u}^h_{xx})_{i-\frac{1}{2}} (v^h)^+_{i-\frac{1}{2}}).\) 
